If you go east from Vermilion City, you'll find Diglett's Cave on Route 11. This cave is no maze; it's simple to go through since there's only one road. It has no items. The only things here that you'll find are Diglett and the occasional Dugtrio. Head inside and go to the other end to go back to Route 2 for some important things.

Route 2

After you exit the cave, you'll revisit Route 2. But this time, it's the other half of the route. Go south, and you'll find a house. Inside, a guy is willing to trade his Mr. Mime for your Abra (Red/Blue), or Clefairy (Yellow).

The next house has one of Professor Oak's Aides in it. If you've collected 10 or more Pokemon he will give you the HM 02 - Flash.